In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple of linear forms in ℤ[x], the set contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps form an increasing (resp. decreasing) sequence. We also show that such strings exist with for 2 ≤ j ≤ m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-4,
author = {William D. Banks and Tristan Freiberg and Caroline L. Turnage-Butterbaugh},
title = {Consecutive primes in tuples},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {261-266},
zbl = {1323.11071},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-4}
}
William D. Banks; Tristan Freiberg; Caroline L. Turnage-Butterbaugh. Consecutive primes in tuples. Acta Arithmetica, Tome 168 (2015) pp. 261-266. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-4/